Accurate prediction of macroscopic transport from microscopic imaging via critical fractals at the Mott transition

TL;DR

The paper tackles the challenge of predicting macroscopic transport in VO$_2$ during the thermally driven metal-insulator transition from spatially resolved optical imaging. By embedding fractal sub-pixel structure into a multiscale resistor-network framework and driving the sub-pixel configuration with a near-critical $2$D RFIM, the authors achieve quantitative matches to the observed $R(T)$ across the full transition range. The approach reveals that fractal domain textures extend down to sub-pixel and potentially unit-cell scales, providing a unified description of microscopic imaging and macroscopic transport with minimal free parameters. This framework not only clarifies VO$_2$ behavior but also offers a transferable method for other fractal-structured materials and neuromorphic applications where optical data can substitute for direct electrical contacts.

Abstract

Vanadium dioxide (VO$_2$) exhibits hysteresis in resistance while undergoing a thermally driven insulator-metal transition (IMT). Understanding the nonequilibrium effects in resistance is of great interest, as VO$_2$ is a strong candidate for brain-inspired computing, which is more energy efficient for AI tasks compared to traditional computing. Accurate models of the connection between microscopic and macroscopic transport properties and microscopic imaging of VO$_2$ will allow us to better utilize VO$_2$ in future applications. However, predictions of macroscopic resistance of VO$_2$ that quantitatively match observations using spatially resolved data have not yet been achieved. Here, we demonstrate an accurate prediction of the macroscopic resistance of VO$_2$ throughout the entire temperature range of interest, by developing a multiscale resistor network model incorporating the assumption of fractal sub-pixel structure of the optical data, where the configuration of insulating and metallic domains within each pixel are drawn from the random field Ising model near criticality. This strongly indicates that the observed fractal, power law structure of metallic and insulating domains extends down to much smaller length scales than the current record for experimental resolution of this system, and that the two-dimensional random field Ising model near criticality is a suitable model for describing the metal and insulator patches of VO$_2$ down to scales that approach the unit cell.

Figures (10)

• Figure 1: Experimental data consisting of: (a) Grayscale optical microscopy images of VO2 at various temperatures. The intensity of each pixel has been normalized for consistency. (b) Experimental data of macroscopic resistance versus temperature. The blue and red lines are linear fits on the log-linear scale where the blue line is the insulating fit and the red line is the metallic fit. The insulating region ($T\leq 40$° C) is fit to an exponential activation $R_{\text{insulating}}(T) = R^0_{\text{insulating}}e^{-AT}$. The metallic region ($T\geq 80$° C) is fit to a constant $R_{\text{metallic}}(T) = R^{0}_{\text{metallic}}$. The orange stars represent the corresponding position on the $R(T)$ curve for the images in panel (a).
• Figure 2: Each plaquette in the figure represents a single pixel from an optical microscopy image. Each pixel is modeled as four resistors coming out of a node (solid black circle), with the value of all four resistors set by the grayscale value of the pixel. In this way, each image is mapped to a unique resistor network. We use $n$ to denote the number of plaquettes parallel to the gold bus bars, and $m$ to denote the number of plaquettes perpendicular to the gold pads. A representative (small) size $n\times m = 3 \times 2$ is shown.
• Figure 3: Mapping random field Ising configurations to macroscopic resistance. (a) Macroscopic resistance vs. average intensity (reported as grayscale) calculated from 4608 $100\times 100$ windows of critical random field Ising configurations. Note the insulating and metallic branches. Resistance is considered insulating if over 10 k$\Omega$ (b) and metallic (c) if less than 10 k$\Omega$. Phenomenological fit to fractal sub-pixel structure predicted by the 2D RFIM, on the insulating branch (d) and metallic branch (e). Data points are put into bins of width 0.01 on the grayscale axis. We obtain the $\braket{R}$ fit curves by fitting to the points computed from the mean value in each bin. The blue region shows one standard deviation within each grayscale bin. Points on the insulating branch are fit down to a grayscale value of 0.3, while points on the metallic branch are fit up to a grayscale value of 0.4. The fitting functions used can be found in Table. \ref{['table:fitting-functions']}, and the parameters are in Table \ref{['table:fitting-parameters-s100']}.
• Figure 4: Schematic of the combined experiment-theory fractal resistor network procedure. Starting from the grayscale experimental surface image (top left), a resistor network was mapped out (top right). Each node (solid black circle) in the resistor network represents one pixel in the image. We compare three different ways to set the value of each resistor: (a) Starting from the experimental image, we use the DOMain INtensity Overturn (DOMINO) method we introduced in Basak2023 to map each pixel to either fully insulating (white pixel) or fully metallic (black pixel). (b) Resistance prediction using the $10\times 10$ fractal sub-pixel fit curve and error band in Table. \ref{['table:fitting-parameters-s10']}. (c) Resistance prediction using the $100\times 100$ fractal sub-pixel fit curve and error band in Fig. \ref{['fig:sub-pixel_s100_combined']}(d) and Fig. \ref{['fig:sub-pixel_s100_combined']}(e). In panels (b) and (c) the grayscale threshold to switch between insulating and metallic branches was set to 0.32.
• Figure 5: Comparison of threshold pixel and fractal RFIM sub-pixel method for macroscopic resistance hysteresis prediction from spatially resolved optical data for the warming branch. $\Delta \log R/\log R_{\rm exp}$ is plotted where $\Delta \log R = \log R_{\rm theory} - \log R_{\rm exp}$. If the error bars cross 0 (denoted by the horizontal dotted line), then the prediction matches within error bars. However, large deviations between the calculated value and the horizontal dotted line indicate lack of agreement between the experimentally derived macroscopic resistance and that predicted theoretically. We can see that the prediction assuming sub-pixel fractal structure gives a much closer match to the data than the raw pixel prediction.